Using a forward log storage and backward log storage to recover a storage to a forward or backward point-in-time

Provided are a computer program product, system, and method for using a forward log storage and backward log storage to recover a storage to a forward or backward point-in-time. In response to receiving writes to source data after establishing point-in-time copies, point-in-time data of the source data is copied to a backward log storage storing point-in-time data for multiple of the point-in-time copies. The point-in-time data in the backward log storage is applied to a recovery source data to roll-back the source data to a backward point-in-time of one of the point-in-time copies. Before applying the point-in-time data from the backward log storage, point-in-time data in the recovery source data, is copied to a forward log storage. The point-in-time data in the forward log storage is applied to the recovery source data to roll forward the recovery source data to a forward point-in-time subsequent to the backward point-in-time.